1-From the first four lessons, you have
learned that in a chaotic system, using the laws of physics to make precise
long-term predictions is impossible, even in theory. Making long-term
predictions to any degree of precision at all would require giving the initial
conditions to infinite precision.

2- At the time of its discovery, the
phenomenon of chaotic motion was considered a mathematical oddity. In the
decades since then, physicists have come to discover that chaotic behavior is
much more widespread, and may even be the norm in the universe.

5- Lorenz's computer code contained the
mathematical equations which governed the flow the air currents. Since computer
code is truly deterministic, Lorentz expected that by inputing the same initial
values, he would get exactly the same result when he ran the program.

9- The mathematics inside Lorenz's model of
atmospheric currents was widely studied in the 1970's. Gradually it came to be
known that even the smallest imaginable discrepancy between two sets of initial
conditions would always result in a huge discrepancy at later or earlier times,
the hallmark of a chaotic system, of course.

10- Scientists now believe that like
Lorenz's simple computer model of air currents, the weather as a whole is a
chaotic system. This means that in order to make long-term weather forecasts
with any degree of accuracy at all, it would be necessary to take an infinite
number of measurements.

11- Even if it were possible to fill the
entire atmosphere of the earth with an enormous array of measuring
instruments---in this case thermometers, wind gauges, and
barometers---uncertainty in the initial conditions would arise from the minute
variations in measured values between each set of instruments in the array.

13- This principle is sometimes called the
"Butterfly Effect." In terms of weather forecasts, the "Butterfly Effect"
refers to the idea that whether or not a butterfly flaps its wings in a certain
part of the world can make the difference in whether or not a storm arises one
year later on the other side of the world.

14- Because of the "Butterfly Effect," it is
now accepted that weather forecasts can be accurate only in the short-term, and
that long-term forecasts, even made with the most sophisticated computer
methods imaginable, will always be no better than guesses.

15- Thus the presence of chaotic systems in
nature seems to place a limit on our ability to apply deterministic physical
laws to predict motions with any degree of certainty. The discovery of chaos
seems to imply that randomness lurks at the core of any deterministic model of
the universe.

16- Because of this fact, some scientists
have begun to question whether or not it is meaningful at all to say that the
universe is deterministic in its behavior. This is an open question which may
be partially answered as science learns more about how chaotic systems operate.

18- Some scientists have speculated that the
presence of chaos---that is, randomness operating through the deterministic
laws of physics on a microscopic level---may actually be necessary for larger
scale physical patterns to arise.

20- As the study of chaos in physics enters
its second century, the issue of whether the universe is truly deterministic is
still an open question, and it will undoubtedly remain so, even as we come to
understand more and more about the dynamics of chaotic systems.